#6816411

order	512
nilpotency class	3
isoclinism	isoclinic class of #6682360
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#469	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1766 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 44 points:

(5,6)(7,11)(8,10)(9,12)(13,14)(15,23)(16,27)(17,19)(18,31)(20,26)(21,28)(22,30)(24,36)(25,39)(29,34)(32,38)(33,44)(35,37)(40,41)(42,43);(2,3)(5,7,8,12)(6,9,10,11)(14,19)(15,24)(21,27)(22,31)(25,37)(26,38)(34,42)(40,44);(1,2,4,3)(6,10)(7,12)(13,15,18,26,17,24,30,38)(14,20,22,23,19,32,31,36)(16,25,29,40,28,37,43,44)(21,33,34,35,27,41,42,39);(6,10)(9,11)(13,16,17,28)(14,21,19,27)(15,25,24,37)(18,29,30,43)(20,33,32,41)(22,34,31,42)(23,35,36,39)(26,40,38,44);(1,3,4,2)(15,24)(16,28)(20,32)(21,27)(23,36)(26,38)(29,43)(34,42)

Polycyclic presentation:

2^9|b^2=fi,c^2=ghi,d^2=i,e^2=fh,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,e^c=ei,e^d=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6723607);